396 Mr. Beverley on the Rectification of Curve Lines. 
cos ? 26, which gives y = asin§. cos? 2, and x =a cos 6. 
ee —(4sin3 — 3sin 4) asin écos? 2¢ 
5 ) . 
cos “26 .*. iy = cosé(1—Asin?#) COs SIN §. 
—sij 6 . 
cos? 24 x — = TD, and a sin§.cos} 26(TD) :1(rad.):: 
asin§cos? 24 (y = CD): —- = — cot36 = — cot TAQ; 
therefore DAQ = 3DAC, and consequently. fAC x d TAQ 
=a cos 2 26x3 d 4, which, by putting cos 3 24= 5 becomes 
2d 
38axf re , the same as from 5, table 3, Landen’s Memoirs 
=? 
(vide Math. Repos. vol. i. page 61, old series) by which the 
whole integral between ¢ = 1 (sin 90°), and ¢ = 0, is readily 
found to be 3a x *5990701173 = 1°7972103519 a .*. the length 
of the four branches is 7°1888414076 a, a being the axis of the 
lemniscata. 
Proposition 2.—AEM is a plane 
curve, AB an indefinite axis; join AM, 
and perpendicular to AB draw AD 
meeting a perpendicular from M in 
D; then if the area AEMA have any 
given ratio to the area AEMD, the 
curve is of the parabolic kind. 
Demonstr.—For put AD =y, MD 
=., and let the ratio of the two areas be as m: n; then we must 
have }AM?x dMAD: MD x dAD:: m:n, or $(2* + y’) x 
ydx—ady 
mty 
; d 
mxdy, and by reduction — 
sPdgne +s) | 28 (24-9). Xx 
of dy ° 
= (2m + n)x ze Taking the 
integrals n x hyp. log a = (2m +n) x hyp. log y, and therefore 
n Qnmin .- 2m n Qnin E 
—— >O°ra #£ =Y > an equation to a curve of the 
. . Qm - . ¥ 
parabolic kind, where a” is an arbitrary constant. _Q.E.D. 
Obs.—When the areas are equal, or m=, it becomes 
a’x = y°, the first cubical parabola. 
When the areas are as 1:2, or m= 1, and 2 = Q, it be- 
comes ax = 7’, the common or Apollonian parabola, &c. &c. 
Iam, sir, your very obedient servant, 
Brompton, near Scarborough, Tuomas BEVERLEY. 
March 6, 1826, ' 
LIX. On 
